The Representation of Facts in Physical Theories

Paul A. Benioff

A quantum theory representations of real (R) and complex (C) numbers is given that is based on states of single, finite strings of qukits for any base k > 1. Both unary representations and the possibility that qukits with k a prime number are elementary and the rest composite are discussed. Cauchy sequences of qukit string states are defined from the arithmetic properties. The representations of R and C, as equivalence classes of these sequences, differ from classical kit string state representations in two ways: the freedom of choice of basis states, and the fact that each quantum theory representation is part of a mathematical structure that is itself based on the real and complex numbers. These aspects enable the description of 3 dimensional frame fields labeled by different k values, different basis or gauge choices, and different iteration stages. The reference frames in the field are based on each R and C representation where each frame contains representations of all physical theories as mathematical structures based on the R and C representation. Approaches to integrating this with physics are described. It is observed that R and C values of physical quantities, matrix elements, etc. which are viewed in a frame as elementary and featureless, are seen in a parent frame as equivalence classes of Cauchy sequences of qukit string states.

Evert Willem Beth

Maria Cristina Abbati, Renzo Cirelli, Franco Gallone

After a schematic examination of the ’’physical’’ representations of the Galilei group, a quantum mechanical description is drawn from a ’’laboratory’’ one in the case of a free elementary system with spin, assuming that the kinematical group is the Galilei group. In fact, the ’’laboratory states’’ of the system naturally fit in a Galilei−Hilbert bundle. Then, according to the general theory of the unitary representations of groups in the framework of Hilbert bundles which is outlined in this paper, a unitary representation of the Galilei group is constructed which is shown to contain all the physical Galilei representations. The quantum mechanical description which has been obtained in this way gives rise to an algebra of observables by means of a general procedure which connects a unitary representation of a Lie group with a Hilbert representation of the corresponding Lie algebra. The inner energy is shown to be a superobservable for the system under consideration. Moreover, the kinematical properties of such a system are independent of the values of the inner energy.

Iulian D. Toader

Mario Bunge

S. Sengupta

A viewpoint is presented which, taken in probability theory and/or in quantum theory, will help place certain facts in a unified perspective and clarify their apparently enigmatic status. These facts are: (1) while the probabilities in probability theory are squared moduli of complex numbers; (2) the measure-theoretic apparatus of probability theory cannot accommodate the quantization rules for physical measurables and, above all; (3) the notion of limits on the accuracy attainable in simultaneous measurements on canonically conjugate variables does not appear in the theorems and algorithms of probability theory. The viewpoint rests on the intuitive but fundamental idea that measurable functions have non-unique inverses, and that an inner product representation of measures can be obtained whenever the function is differentiable. This idea is generalized to functions measurable on real rectangles and on complex plane. The formulas derived for measures are formally the same as those in quantum mechanics. The remainder of the development is based on the notion of a measurable-preserving point-to-point mapping from the range of these special measurable functions to an appropriate subset of a Hilbert space. Physical interpretations are attempted for each of the measure-theoretic constructs.